Math. Z. https://doi.org/10.1007/s00209-018-2094-1 Mathematische Zeitschrift Property P for acylindrically hyperbolic groups naive 1 2 Carolyn R. Abbott · François Dahmani Received: 2 September 2017 / Accepted: 13 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract We prove that every acylindrically hyperbolic group that has no non-trivial ﬁnite normal subgroup satisﬁes a strong ping pong property, the P property: for any ﬁnite nai ve collection of elements h ,..., h , there exists another element γ = 1 such that for all 1 k i, h ,γ=h ∗γ . We also show that if a collection of subgroups H ,..., H is a i i 1 k hyperbolically embedded collection, then there is γ = 1 such that for all i, H ,γ = H ∗γ . i i Keywords Acylindrically hyperbolic groups · δ-hyperbolic spaces · C -algebras · Free subgroups · Ping-pong lemma · Property P nai ve Mathematics Subject Classiﬁcation 20E07 · 20F65 · 46L35 The Ping-Pong lemma and its many variations are iconic arguments in group theory that are particularly useful when dealing with groups acting on hyperbolic spaces. They allow one to produce free subgroups at will in many groups.
Mathematische Zeitschrift – Springer Journals
Published: Jun 5, 2018
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